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In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\R^2\times H^1(\R^3,\R^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\R^3\\ -\Delta v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\R^3\\ u,v>0&\hbox{in}\;\R^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\R^3}u^2=a^2\quad\hbox{and}\;\int_{\R^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the case of fixed frequencies $\lambda_1,\lambda_2$. When the masses are prescribed, the standard approach to this problem is variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $\beta$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist for all $\beta>0$ and all $a,b>0$.

We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochastic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Green’s function, we show that either the deterministic or the random part of the corrector dominates.

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.